Question: Given $ m \angle QPR = 5x + 49$, and $ m \angle RPS = 4x + 122$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since $\angle QPS$ is a straight angle, we know ${m\angle QPS = 180}$ Substitute in the expressions that were given for each measure: $ {5x + 49} + {4x + 122} = {180}$ Combine like terms: $ 9x + 171 = 180$ Subtract $171$ from both sides: $ 9x = 9$ Divide both sides by $9$ to find $x$ $ x = 1$ Substitute $1$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 4({1}) + 122$ Simplify: $ {m\angle RPS = 4 + 122}$ So ${m\angle RPS = 126}$.